I have a clock at home that chimes on every hour and half hour mark. On whole hour marks it chimes once for each hour, and on half hour marks it chimes only once.

One night I was wakened suddenly and realized the the clock had chimed, but I did not know how many times. As I lay awake thinking about nothing in particular I heard the clock chime once and started wondering what time it was.

What is the longest amount of time I would have to lie awake before I knew for sure what time it was?

Once you've heard three single chimes in a row, at 1:30, you'd know it was 1:30, since this is the only time that there is in fact a third single chime in a row. You would not have to wait till 2:00 to determine the time. From 12:00 to 1:30 is only 1 1/2 hours.